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Mirrors > Home > MPE Home > Th. List > funcnv0 | Structured version Visualization version GIF version |
Description: The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.) |
Ref | Expression |
---|---|
funcnv0 | ⊢ Fun ◡∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fun0 6633 | . 2 ⊢ Fun ∅ | |
2 | cnv0 6163 | . . 3 ⊢ ◡∅ = ∅ | |
3 | 2 | funeqi 6589 | . 2 ⊢ (Fun ◡∅ ↔ Fun ∅) |
4 | 1, 3 | mpbir 231 | 1 ⊢ Fun ◡∅ |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 4339 ◡ccnv 5688 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-fun 6565 |
This theorem is referenced by: f10 6882 pthdlem1 29799 0trl 30151 0pth 30154 |
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